# The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.

Let a and r be the first term and the common ratio of the G.P. respectively.

a1 = 1

nth term of G.P is given by -

an = arn - 1

a3 = ar2 = (1)r2

a6 = ar4 = (1)r4

According to question -

a3 + a6 = 90

r2 + r4 = 90

r4 + r2 – 90 = 0

Assume r2 = t

t2 + t – 90 = 0

t2 + 10t - 9t – 90 = 0

t(t + 10) - 9(t + 10) = 0

(t - 9)(t + 10) = 0

t =9 or t = - 10(invalid because t = r2, so t can't be - ve.)

r2 = 9 ( Taking real roots)

r = 3

Thus, the common ratio of the G.P. is 3.

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